A second order \(\eta \)-approximation method for constrained optimization problems involving second order invex functions. (English) Zbl 1212.90307

Antczak, Tadeusz

Appl. Math., Praha 54, No. 5, 433-445 (2009).

Summary: A new approach for obtaining second order sufficient conditions for nonlinear mathematical programming problems which makes use of the second order derivative is presented. In the so-called second order \(\eta \)-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order \(\eta \)-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second order \(\eta \)-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.

MSC:

90C26	Nonconvex programming, global optimization
90C30	Nonlinear programming
90C46	Optimality conditions and duality in mathematical programming

Keywords:

second order \(\eta \)-approximated optimization problem; second order invex function; second order optimality conditions

References:

[1]	T. Antczak: An {\(\eta\)}-approximation approach to nonlinear mathematical programming problems involving invex functions. Numer. Funct. Anal. Optimization 25 (2004), 423–438. · Zbl 1071.90032
[2]	M. S. Bazaraa, H. D. Sherali, C. M. Shetty: Nonlinear Programming. Theory and Algorithms. John Wiley & Sons, New York, 1993.
[3]	C. R. Bector, B. K. Bector: (Generalized)-bonvex functions and second order duality for a nonlinear programming problem. Congr. Numerantium 52 (1985), 37–52.
[4]	C. R. Bector, M. K. Bector: On various duality theorems for second order duality in nonlinear programming. Cah. Cent. Etud. Rech. Opér. 28 (1986), 283–292. · Zbl 0622.90068
[5]	C. R. Bector, S. Chandra: Generalized bonvex functions and second order duality in mathematical programming. Res. Rep. 85-2. Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, 1985.
[6]	C. R. Bector, S. Chandra: (Generalized) bonvexity and higher order duality for fractional programming. Opsearch 24 (1987), 143–154. · Zbl 0638.90095
[7]	A. Ben-Israel, B. Mond: What is invexity? J. Aust. Math. Soc. Ser. B 28 (1986), 1–9. · Zbl 0603.90119
[8]	A. Ben-Tal: Second-order and related extremality conditions in nonlinear programming. J. Optimization Theory Appl. 31 (1980), 143–165. · Zbl 0416.90062
[9]	B. D. Craven: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24 (1981), 357–366. · Zbl 0452.90066
[10]	M. A. Hanson: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), 545–550. · Zbl 0463.90080
[11]	O. L. Mangasarian: Nonlinear Programming. McGraw-Hill, New York, 1969.
[12]	D. H. Martin: The essence of invexity. J. Optimization Theory Appl. 47 (1985), 65–76. · Zbl 0552.90077